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Search: id:A054520
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| A054520 |
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Let S = {1,5,9,13,..., 4n+1, ...} and call p in S an S-prime if p>1 and the only divisors of p in S are 1 and p; sequence gives elements of S that are not S-primes. |
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4
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| 1, 25, 45, 65, 81, 85, 105, 117, 125, 145, 153, 165, 169, 185, 189, 205, 221, 225, 245, 261, 265, 273, 285, 289, 297, 305, 325, 333, 345, 357, 365, 369, 377, 385, 405, 425, 429, 441, 445, 465, 477, 481, 485, 493, 505, 513, 525, 533, 545, 549, 561, 565, 585
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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The set S is a standard example of a set where unique factorization does not hold.
With the exception A054520(1)=1, numbers of the form 4*(m + n + 4 m n)+1 (m,n>0). No such number can be prime because 4*(m + n + 4 m n)+1=(4m+1)(4n+1) [From Artur Jasinski (grafix(AT)csl.pl), Sep 22 2008]
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 101, problem 1.
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LINKS
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Eric Weisstein's World of Mathematics, Hilbert Number [From Eric W. Weisstein (eric(AT)weisstein.com), Sep 15 2008]
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EXAMPLE
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49 is an S-prime.
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MATHEMATICA
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a = {}; Do[Do[AppendTo[a, 4(m + n + 4 m n)+1], {m, 1, 100}], {n, 1, 100}]; Union[a] [From Artur Jasinski (grafix(AT)csl.pl), Sep 22 2008]
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CROSSREFS
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Cf. A057948, A057949, A057950.
Sequence in context: A138091 A105507 A015911 this_sequence A038811 A028505 A154082
Adjacent sequences: A054517 A054518 A054519 this_sequence A054521 A054522 A054523
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Apr 09 2000
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 11 2000
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